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Συναλλοιότητα Lorentz
Συναλλοιότης Lorentz Lorentz covariance thumb|300px| [[Συναλλοιότητα Lorentz ]] - Μία ιδιότητα Ετυμολογία Η ονομασία "Συναλλοιότητα" σχετίζεται ετυμολογικά με την λέξη "αλλοίωση". Εισαγωγή In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings: 1. A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a "Lorentz invariant" (i.e., they transform under the trivial representation). 2. An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term "invariant" here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity, i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference. This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities. Examples In general, the nature of a Lorentz tensor can be identified by its tensor order, which is the number of free indices it has. No indices implies it is a scalar, one implies that it is a vector, etc. Furthermore, any number of new scalars, vectors etc. can be made by contracting or creating an outer product of any kinds of tensors together, but many of these may not have any real physical meaning. Some of those tensors that do have a physical interpretation are listed (by no means exhaustively) below. Please note, the metric sign convention such that η = diag (1, −1, −1, −1) is used throughout the article. Scalars Spacetime interval: : \Delta s^2=\Delta x^a \Delta x^b \eta_{ab}=c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 Proper time (for timelike intervals): : \Delta \tau = \sqrt{\frac{\Delta s^2}{c^2}},\, \Delta s^2 > 0 Proper distance (for spacelike intervals): : L = \sqrt{-\Delta s^2},\, \Delta s^2 < 0 Rest mass: : m_0^2 c^2 = P^a P^b \eta_{ab}= \frac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2 Electromagnetism invariants: : F_{ab} F^{ab} = \ 2 \left( B^2 - \frac{E^2}{c^2} \right) : G_{cd}F^{cd}=\frac{1}{2}\epsilon_{abcd}F^{ab} F^{cd} = - \frac{4}{c} \left( \vec B \cdot \vec E \right) D'Alembertian/wave operator: : \Box = \eta^{\mu\nu}\partial_\mu \partial_\nu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2} Four-vectors 4-Displacement: : \Delta X^a = (c\Delta t, \vec{\Delta x}) = (c\Delta t, \Delta x, \Delta y, \Delta z) 4-Position: : X^a = (ct, \vec{x})= (ct, x, y, z) 4-Gradient: with is the 4D Partial derivative: : \partial^a = \left(\frac{\partial_t}{c}, -\vec{\nabla}\right) = \left( \frac{1}{c}\frac{\partial}{\partial t}, -\frac{\partial}{\partial x}, -\frac{\partial}{\partial y}, -\frac{\partial}{\partial z} \right) 4-Velocity: : U^a = \gamma(c,\vec{u}) = \gamma \left(c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) where U^a = \frac{dX^a}{d\tau} 4-Momentum: : P^a = (mc,\vec{p}) = \left(\frac{E}{c},\vec{p}\right)= \left(\frac{E}{c}, p_x, p_y, p_z\right) where P^a = m_o U^a 4-Current: : J^a = (c\rho,\vec{j}) = \left(c\rho, j_x, j_y, j_z\right) where J^a = \rho_o U^a Four-tensors The Kronecker delta: : \delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \\ 0 & \mbox{if } a \ne b. \end{cases} The Minkowski metric (the metric of flat space according to general relativity): : \eta_{ab} = \eta^{ab} = \begin{cases} 1 & \mbox{if } a = b = 0, \\ -1 & \mbox{if }a = b = 1, 2, 3, \\ 0 & \mbox{if } a \ne b. \end{cases} The Levi-Civita symbol: : \epsilon_{abcd} = -\epsilon^{abcd} = \begin{cases} +1 & \mbox{if } \{abcd\} \mbox{ is an even permutation of } \{0123\}, \\ -1 & \mbox{if } \{abcd\} \mbox{ is an odd permutation of } \{0123\}, \\ 0 & \mbox{otherwise.} \end{cases} Electromagnetic field tensor (using a metric signature of + − − − ): : F_{ab} = \begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c \\ -E_x/c & 0 & -B_z & B_y \\ -E_y/c & B_z & 0 & -B_x \\ -E_z/c & -B_y & B_x & 0 \end{bmatrix} Dual electromagnetic field tensor: : G_{cd} = \frac{1}{2}\epsilon_{abcd}F^{ab} = \begin{bmatrix} 0 & B_x & B_y & B_z \\ -B_x & 0 & E_z/c & -E_y/c \\ -B_y & -E_z/c & 0 & E_x/c \\ -B_z & E_y/c & -E_x/c & 0 \end{bmatrix} Υποσημειώσεις Εσωτερική Αρθρογραφία *ανταλλοιότητα *αναλλοιότητα, ανταλλοιότητα * παγκοσμιότητα, τοπικότητα * Coordinate conditions * Coordinate-free * Covariance and contravariance * Covariant derivative * Diffeomorphism * Fictitious force * Galilean invariance * Gauge covariant derivative * General covariant transformations * Harmonic coordinate condition * Inertial frame of reference * Lorentz covariance * Principle of covariance * Special relativity * Symmetry in physics Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *Συναλλοιότητα Μπαντές *[ ] Κατηγορία:Σχετικιστική Φυσική